Get married: act vs ceremony

You can use marriage to refer to the act of getting married, Her family did not approve of her marriage to David.

You don't usually use marriage to refer to the ceremony in which two people get married. Use wedding.

Marriage | WordReference.com

I don't understand the purported difference in meaning between the act and the ceremony of getting married.

Maybe using wedding here would be ambiguous, with a possible interpretation "they didn't like the wedding planner's whole organization"?

According to Oxford Advanced Learner's Dictionary, marriage clearly can be used to refer to the ceremony.

marriage [noun] ...

(3) [countable] the ceremony in which two people marry each other

Their marriage took place in a local church.

Though OALD adds the caveat:

Wedding is more common in this meaning.

The following provides a good example where 'wedding' and 'marriage' are non-interchangeable:

• Their marriage was a disaster implies that the couple were not happy in their life together and are probably now separated or divorced.

On the other hand,

• Their wedding was a disaster implies that something happened during the wedding and that the ceremony did not go smoothly. The couple can go on to have a long and happy marriage even after a disastrous wedding.